Set-theoretic limit

For a sequence of sets, the limit superior of sets consists of points belonging to infinitely many members, while the lower limit consists of points belonging to every sufficiently late member. If the two sets coincide, their common value is the set-theoretic limit.

Membership means that for every index $N$ there is a later $n\ge N$ with $x\in E_n$, equivalently membership for infinitely many indices. If ${\sum }_n\mu (E_n)<\infty$, countable subadditivity gives $\mu (\mathrm{limsup}{E}_n)\le {\sum }_{n\ge N}\mu (E_n)\to 0$. This is the measure form of the first Borel-Cantelli lemma, and requires no independence assumption.